Forward leakage current, look at the curve for Vf versus current, and you will see there is considerable current at low voltage, and the zener voltage is specified at a certain current. Even worse this current varies with both temperature and time, and also with stray light hitting the junction. Low voltage zener diodes are very poor, generally the application has them run at a specified current, typically 5 to 50mA, as per datasheet, and you only will get the voltage there, higher current higher voltage, and you can also look at slope resistance to predict the voltage change.
You want to use it to protect you run it from a higher voltage via a resistor, and the use a low leakage diode to switch current from input overload into it, you typically use the BC junction of a bipolar transistor, as they have low leakage, and normally a 30V reverse voltage. Need 2, one for positive peaks, and one from negative, or use another transistor to provide that clamp, and accept the increased power dissipation in the zener clamp.